Linear Sequences with Non-Constant Transition Dynamic Programming
The "Linear Sequences with Non-Constant Transition" dynamic programming pattern is used to solve problems where the DP problem is solved on every prefix of the array, similar to the "Group 2" pattern. However, the transitions in this pattern may not be simple and may require considering a linear amount of options from indices \(j < i\).
In this pattern, we still consider solving the DP problem on every prefix of the array. However, the transition from one state to another may not be constant and may depend on a linear number of options from previous indices.
Top-Down Approach:
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12 | def dp_top_down(i):
if i == 0:
return base_case_value
if dp[i] is already computed:
return dp[i]
# Initialize with a value that works for the base case
dp[i] = initial_value
# Iterate over possible transitions from indices j < i
for j in range(i-1, -1, -1):
# Update dp[i] based on transitions from previous indices
dp[i] = max(dp[i], some_function(dp[j], ...))
return dp[i]
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Bottom-Up Approach:
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12 | def dp_bottom_up():
# Initialize base cases
dp[0] = base_case_value
# Fill DP array
for i in range(1, n):
# Initialize with a value that works for the base case
dp[i] = initial_value
# Iterate over possible transitions from indices j < i
for j in range(i-1, -1, -1):
# Update dp[i] based on transitions from previous indices
dp[i] = max(dp[i], some_function(dp[j], ...))
return max(dp) # Final answer
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